Transition to parametric wave patterns in a vertically oscillated granular layer

Experiments on vertically oscillated granular layers reveal, at a critical acceleration, a well-defined transition from a flat surface to standing wave patterns oscillating at half the excitation frequency. The patterns observed in a cylindrical container are squares or stripes; a continuous transition between these states occurs when the vibration frequency is varied at constant acceleration. The dispersion relation for these parametric granular waves is similar to that for gravity waves in a fluid, but exhibits a well-defined cutoff associated with the particle size.